Out of 18 points in a plane no three are in the same straight line except five points which are collinear . The number of

Options

A. 140
B. 142
C. 144
D. 146

Answer

To solve the problem, we need to determine the number of straight lines that can be formed from 18 points in a plane, where no three points are collinear except for 5 points that are collinear.

### Step-by-Step Solution:

1. **Identify the total points and collinear points:**
- Total points (n) = 18
- Collinear points (p) = 5

2. **Calculate the total number of lines formed by choosing any 2 points from the 18 points:**
- The number of ways to choose 2 points from 18 is given by the combination formula:
\[ \binom{n}{2} = \frac{n(n-1)}{2} \]
- So, we calculate:
\[ \binom{18}{2} = \frac{18 \times 17}{2} = 153 \]

3. **Calculate the number of lines formed by the collinear points:**
- Since 5 points are collinear, they will only form 1 line instead of the 10 lines that would be formed by choosing any 2 points from these 5 points.
- The number of ways to choose 2 points from the 5 collinear points is:
\[ \binom{5}{2} = \frac{5 \times 4}{2} = 10 \]
- However, all these 10 combinations give the same line, so we only count this as 1 line.

4. **Adjust the total number of lines:**
- We need to subtract the extra lines counted from the collinear points:
\[ \text{Total lines} = \text{Lines from all points} - \text{Extra lines from collinear points} + 1 \]
- Thus:
\[ \text{Total lines} = 153 - 10 + 1 = 144 \]

5. **Final Answer:**
- The total number of straight lines that can be formed is **144**.